The answer has to do with how many circles are in each number. So the number 8 has two circles in its shape so it counts as two. And 0 is one big circle, so it counts as 1. So 2581=2. Ok, that’s cute, it’s an alternative mapping of values with implied addition.

What bugged me was how might I solve this if the mapping of values was not based on shape. So how could I program a computer to solve this puzzle? I gave it a little thought and since I like to pretend I’m an econometrician, this looked a LOT like a series of equations that could be solved with an OLS regression. So how can I refactor the problem and data into a trivial OLS? I really need to convert each row of the training data into a frequency of occurrence chart. So instead of 8809=6 I need to refactor that into something like:

1,0,0,0,0,0,0,0,2,1 = 6

In this format the independent variables are the digits 0-9 and their value is the number of times they occur in each row of the training data. I couldn’t figure out how to do the freq table so, as is my custom, I created a concise simplification of the problem and put it on StackOverflow.com which yielded a great solution. Once I had the frequency table built, it was simple a matter of a linear regression with 10 independent variables and a dependent with no intercept term.

My whole script, which you should be able to cut and paste into R, if you are so inclined, is the following:

## read in the training data## more lines than it should be because of the https requirement in Github
temporaryFile <-tempfile()download.file("https://raw.github.com/gist/2061284/44a4dc9b304249e7ab3add86bc245b6be64d2cdd/problem.csv",destfile=temporaryFile, method="curl")
series <-read.csv(temporaryFile)## munge the data to create a frequency table
freqTable <-as.data.frame(t(apply(series[,1:4],1,function(X)table(c(X,0:9))-1)))names(freqTable)<-c("zero","one","two","three","four","five","six","seven","eight","nine")
freqTable$dep <- series[,5]## now a simple OLS regression with no intercept
myModel <-lm(dep ~0+ zero + one + two + three + four + five + six + seven + eight + nine,data=freqTable)round(myModel$coefficients)